The visitors had by far the better of the chances, particularly in the first period, and Atletico were grateful to goalkeeper Jan Oblak for keeping them in the tie with a string of fine saves.
Tempers flared after the break as Madrid defender Dani Carvajal escaped unpunished after punching Mario Mandzukic inside the penalty area.
However, for the fourth time this season Real failed to score at the Calderon as Atletico extended their unbeaten run in the Madrid derby this season to seven games.
And the sides meet for the eighth time at the Santiago Bernabeu on April 22.
"Obviously you always want to score and win, but it is important not to have not conceded at home," Oblak told Television Espanola.
"I don't know (who is the favourite now). Madrid have a great team with a lot of great players so we will see what happens in the return leg."
Real boss Carlo Ancelotti, meanwhile, was left to lament his side not killing the tie off before half-time.
"I think that in the first-half we deserved more than to be 0-0 at half-time. We need to highlight the performance in the first-half which was fantastic.
"In the second-half, Atletico pressured more and the game was more even.
"It isn't a brilliant result but we have confidence for the return leg, especially based on the first-half performance."
Atletico coach Diego Simeone kept faith with Oblak in goal ahead of the fit again Miguel Angel Moya and that decision paid dividends after just four minutes when Gareth Bale pounced on an uncharacteristic error from Diego Godin, but placed his shot too close to the on-rushing Slovenian, who spread himself well to make the save.
"The tie is still open," said Simeone.
"In the first half the opponents were good. Oblak was great, especially in the chance for Bale which was the clearest opportunity.
"The best thing was the reaction we had. Any other team that had been damaged as we were in the first half would have ended up relenting, but in the second-half the game became much closer to what we had imagined at the start."
Oblak was kept busy throughout the first 45 minutes as he held a free-kick from Cristiano Ronaldo before palming away another fiercely struck shot from Bale.
His best stop was yet to come as he dived low to his left to turn James Rodriguez's inventive attempt with the outside of his left foot to safety.
The chances continued to come for the visitors as Luka Modric blasted just over when well-placed at the edge of the box before Oblak was tested again by Rodriguez at the end of a stunning counter-attack led by Raphael Varane.
Atletico started the second period more brightly with Arda Turan heading wide from Juanfran's cross.
Mandzukic was then left with a bloody nose thanks to a stray elbow from Sergio Ramos and the Croatian was on the end of more rough treatment soon after when Carvajal punched him in the stomach.
Carvajal was extremely fortunate not to concede a penalty, but given he wasn't punished at the time by Serbian referee Milorad Mazic he could yet be banned for the return leg.
Real still dominated possession after the interval, but couldn't get the break of the ball inside the area as Atletico somehow scrambled the ball clear 20 minutes from time with Ronaldo and Bale having numerous attempts blocked.
The European champions were dealt a blow a minute from time as Marcelo saw a yellow card that will keep him out of the second leg.
And it was Real who were forced to hold on in the final stages as Iker Casillas scrambled clear Mario Suarez's effort from close range.
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